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Kind of outside the realm of the discussion but I was an active .22 target shooter for many years and the habits ingrained there, where shooting with both eyes open and focusing the the front sight were kind of at odds with each other, transferred over to handgun shooting and I have found myself closing my non-shooting eye despite my best efforts, even shooting IDPA.

My solution when target shooting was to wear a blinder on my non-shooting eye, which kind of does away with the peripheral vision issue.

As far as focusing on the front sight, the sights I was used to in target shooting were "Globe" sights with aperture rear sights. The effect was that, for bullseye shooting, you were looking through a circle, at a circle, around a circle. The human eye very naturally tries to align the circles, so focusing on the front sight gave you a sharp sight, surrounded by a fuzzy circle, around a fuzzy circle, with the eye's natural tendency to align the circles and leads to really good accuracy. This does not as readily transfer to IDPA, USPSA, and such, although front sight focus is still the best way to go.